This site is the blogging component for my main site Crank Astronomy (formerly "Dealing with Creationism in Astronomy"). It will provide a more interactive component for discussion of the main site content. I will also use this blog to comment on work in progress for the main site, news events, and other pseudoscience-related issues.

Here we'll explore some more details about the two-body problem using Newton's laws and gravitation. An n-body code solves Newton's equations by computing the changing positions of multiple objects subjected to various forces. In this case, the forces used are the gravitational forces between the two bodies computed by Newton's Law of Gravitation:

Here M1 & M2 are the two masses and r12 is the distance between the. 'G' is the Newtonian gravitational constant. I provided movies of a simple 2-body problem in my first post on the n-body code (Doin' Real Science: Simulating Particles).

In this post, I'll examine the 2-body problem in more detail.

Setup

For these types of problems in science, we must first setup *Initial conditions* for the problem. The program will then compute the motions forward in time, updating velocities and positions based on the forces on the objects (in this case, gravity).

Let's start with two simple objects, which we call Primary and Secondary,

I'll specify the object speeds in astronomical units per year (AU/yr) and the masses relative to the mass of the Sun. To convert the speed to more familiar units:

1 au/yr = (149.598e6 km)/(365.25*24*3600 s) = 4.74 km/s

= (92.955e6 miles/(365.25*24 hr) = 10604 miles per hour

So to convert my speeds to more familiar values, multiply the AU/year value by 4.74 to get kilometers/second or 10604 to get miles/hour or 2.9455 to get miles per second.

In the computer program, I give the Primary and Secondary objects initial positions and velocities consistent with this setup. We call them INITIAL positions and velocities because they will change with time as the force acts on them. For simplicity, I place the Primary object at the origin (0, 0, 0), the *center* of our universe as we have defined it, and give it an initial velocity of zero (0, 0, 0).

I place the Secondary object at (1AU, 0, 0) and give it a velocity in the y-direction (perpendicular to the initial line between the Primary & Secondary, which defines the x-axis). For this case, I set the magnitude of the velocity to 60% of the speed of a circular orbit at the distance of 1AU.

To make the problem more general, I add an additional velocity of 1AU/yr in the y-direction for both of these objects. Here's a printout of the table of initial parameters:

The position vector is 'x' and the velocity vector is 'v'. The 'Q' column specifies that the planets have no significant electric charge.

Analytic Solution

Using Kepler's laws, we can compute a few specifics about our orbit before running the simulation. This will help provide a check on the accuracy of the simulation run. N-body problems accumulate small errors at each step of the computation so as they run for longer and longer times, their accuracy declines. Here's some Keplerian parameters based on the input values.

Because we chose a speed for the Secondary object that is less than the circular orbit velocity at 1AU, we see that our initial position will correspond to the apoapsis (or farthest distance) of our orbit from the Primary object. When we start running the simulation, we expect the Secondary object to fall *towards* the Primary object.

Numerical Solution

So we run the simulation in the coordinate system as we have defined. Here's a plot of the output dataset.

The red curve tracks the motion of the Primary and the green curve tracks the motion of the Secondary. The black dashed curve corresponds to the motion of the center of mass of the two bodies, computed at each time we've plotted.

We note several things about the motion

the primary object, while it started out at the origin with zero velocity, did not stay at the origin. This is a problem for Geocentrists since it will happen for all objects moving under mutual forces.

both masses move around the center of mass point, the CM lying on the line between them. These are the motions measured when searching for planets by the radial velocity method (wikipedia).

Remember that I gave both objects an initial motion of 1AU/yr in the y-direction. Yet the CM is moving slightly faster, 1.34 AU/yr. This is again due to the fact that gravitation is a mutual force and Newton's laws require an action & reaction.

But those unfamiliar with these types of problems, these paths just look like wavy lines. WHERE IS THE ORBIT?

Kepler's laws are actually defined in the reference frame were one of the objects is at rest, as I noted in the earlier post. So we must compute the RELATIVE positions of the objects. Here we do the case of the motion relative to the Primary (more massive) body.

x(1,ref), y(1,ref), z(1,ref) = position of object 1, in the coordinate system where the origin is object ref.

For the Primary body, this becomes:

x(1,ref) = x(1) - x(1) = 0.0
y(1,ref) = y(1) - y(1) = 0.0

So the Primary resides at the origin in this new coordinate system. Similarly, we compute the new position of the Secondary object at each time step:

With this plot, we can more easily compare the plot to the computed orbital period, semi-minor axis, and periapsis in the table above with the aid of the grid overlay. The secondary (green) completes the orbit between t=0.440 yr and t=0.441 yr, consistent with P=0.4407 yr computed from the input values. We also observe the speed-up of the Secondary as it approaches the Primary mass, as predicted by Kepler's Laws. I also plot the center-of-mass (black dashed line) in this reference frame. In this example, the CM moves in an ellipse around the primary mass.

Other Frames of Reference

Because mathematically, the frames are equivalent, we can also convert to the frame of the less massive object as well!

1. The orbital mechanics of Newton dictates the earth orbits the sun’s
center of mass in an ellipse, yet Newtonian mechanics states the earth
also orbits the solar system barycenter. As the solar system barycenter
is almost never at the center of mass of the sun, then the earth simply
cannot be orbiting the sun in an ellipse. Therefore Newton’s principle
of barycentric motion invalidates Kepler’s laws of elliptical motion.

But now we can see these motions are perfectly compatible. The only evidence Mr. Martin presents for his claim is his 'say so'.

In fact, I can also plot the motion in the original input frame (green & red) and overlay the motion of the secondary mass relative to the primary mass (magenta).

Notice that from the input frame, the Keplerian orbit is not fixed in space, but appears to be carried by the primary mass as the primary moves around the center of mass.

N-body codes are a regular tool for astronomy and celestial navigation. While I don't include many of the smaller forces (planetary oblateness, relativisitic effects, etc.) my tool is sufficiently accurate to demonstrate many principles in the application of Newton's laws with gravitation.

I've got a number of simulations of Geocentrists' claimed "problems" that I'm writing up for future posts.

3 comments:

Tom - the primary object, while it started out at the origin with zero velocity, did not stay at the origin. This is a problem for Geocentrists since it will happen for all objects moving under mutual forces.

JM – It’s only a problem for geocentrists if we believe there are only two bodies in the universe Tom. We geocentrists believe Newtonian mechanics is fundamentally flawed anyway, so using a flawed model only produces flawed apparent problems for geocentrists which really don’t exist at all.

Tom - both masses move around the center of mass point, the CM lying on the line between them. These are the motions measured when searching for planets by the radial velocity method (wikipedia).

JM – CM is merely a maths figment that only exists in the minds of Newtonians and their models. It doesn’t exist in the real because mass attraction is a myth.

Tom - Remember that I gave both objects an initial motion of 1AU/yr in the y-direction. Yet the CM is moving slightly faster, 1.34 AU/yr. This is again due to the fact that gravitation is a mutual force and Newton's laws require an action & reaction.

JM – Newton’s laws also require fictional forces to be added in, in certain inertial frame, because the forces don’t exist in the model. So what are we to do with Newtonian mechanics? Do we admit it is of only limited quantitative value or do we over exaggerate the power of the model, even though geocentrists have said what most physicists already know about Newtonian mechanics – it’s a model based upon assumptions, geometry and maths and a small amount of physical experiment to determine some constants.

Its also a model that does not correctly model the motions of galaxies and that why Newtonian mechanics has been modified to MOND http://en.wikipedia.org/wiki/Modified_Newtonian_dynamics

Evidently, even with this large problem within the model, Tom thinks Newtonian mechanics still represents the real motion of many objects.

Some questions for Tom –

Newtonian mechanics proposes absolute space as a starting assumption within the model. What does modern science have to say experimentally concerning the existence of absolute space? How does absolute space stack up against the relative space f relativity?

Newtonian mechanics proposes action at a distance, so how is this action known to exist experimentally? How does action at a distance occur under relativity at c, when there is no upper limit in Newtonian mechanics?

Tom quotes JM after Tom has shown us the results of a two body problem –

1. The orbital mechanics of Newton dictates the earth orbits the sun’s center of mass in an ellipse, yet Newtonian mechanics states the earth also orbits the solar system barycenter. As the solar system barycenter is almost never at the center of mass of the sun, then the earth simply cannot be orbiting the sun in an ellipse. Therefore Newton’s principle of barycentric motion invalidates Kepler’s laws of elliptical motion.

Tom - But now we can see these motions are perfectly compatible. The only evidence Mr. Martin presents for his claim is his 'say so'.

JM – So Tom, my statement is concerned with the inconsistency between the barycenters required for the two body and n body problems. In the two body problem, the earth orbits the barycenter near the center of the sun. In the n body problem the earth orbits the CM of the solar system, which is usually located outside the sun. so Newtonian mechanics is inconsistent with Keplers laws and with itself. Why?

Newtonian mechanics says the earth orbits the CM of the earth-sun system which is located near the sun’s center.

Newtonian mechanics says the earth orbits the CM of the earth – solar system barycenter.

The two barycenters are rarely at the same point in space, so the theory is self contradictory.

Likewise Keplers laws say the earth orbits in space as an ellipse around a barycenter. So this barycenter is then at the center of the sun and not the center of the sun according to Newtonian mechanics. Therefore Newtonian mechanics is both self contradictory and contradicts Kepler’s laws.

Tom - Notice that from the input frame, the Keplerian orbit is not fixed in space, but appears to be carried by the primary mass as the primary moves around the center of mass.

JM – Tom’s efforts are all misdirected of course, because my objection to Newtonian mechanics requires an interaction between the 2 body and n body problems. That fact is that when we introduce the n body problem it is utterly impossible for bodies to have elliptical orbital paths due to the ever moving barycenter of the n body system.

Once again, Newtonain mechanics is easily shown to be off only limited modeling value and does not propose an serious problems for geocentrism.

1) Kepler's laws in cases of more than two bodies are, at best, approximate,. We see an example of the 3-body case in the later post, Geocentrist 3-Body 'Problem'. Both the Sun, "Earth", and "Jupiter" have no problem moving around the common CM of the system.

2) Newtonian mechanics does not require absolute space, as demonstrated in Geocentrism vs. Inertial Frames. Newtonian mechanics only requires an absolute time. Newton's laws work just fine in any inertial frame which differ in position and velocity.

3) Inertial frames do not have to deal with 'fictitious forces' (wikipedia). This is only an issue for non-inertial frames (accelerating or rotating). GR eliminates fictitious forces by incorporating them in the metric.

4) "Action at a Distance", while a 'problem' is not a show stopper. "Action at a distance" happens in electromagnetism as well. We used the equations of fluid mechanics and gas dynamics long before we understood they were due to atoms. Not fully understanding how the mechanism works is not necessarily a problem when it comes to using the mathematical relationships describing the interaction.

The equations still work.

The fact that Mr. Martin is using a computer is proof that the math works. There is no telling how many other technologies Mr. Martin enjoys that rely on science he has chosen not to believe. It would be interesting to develop a more definitive test of Mr. Martin's understanding.

Science works even if you don't believe it.

Of course, if you choose to believe the WRONG science, you can pay a pretty high price if you make decisions based on the errors (Whatstheharm.net)

"The point is that we are all capable of believing things which we know to be untrue, and then, when we are finally proved wrong, impudently twisting the facts so as to show that we were right. Intellectually, it is possible to carry on this process for an indefinite time: the only check on it is that sooner or later a false belief bumps up against solid reality, usually on a battlefield."-- George Orwell, 1946

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About Me

I obtained my doctorate in physics and astronomy in 1994. I currently work in scientific data visualization for the media and public outreach. For more information on how I became involved in the creationism issue, visit my main page